Disjoint matchings of graphs

Abstract

In this paper, we prove a generalization of the familiar marriage theorem. One way of stating the marriage theorem is: Let G be a bipartite graph, with parts S 1 and S 2. If A ⊂ S 1 and F( A) ⊂ S 2 is the set of neighbors of points in A, then a matching of G exists if and only if Σ x∈ S 2 min(1, | F −1( x) ∩ A |) ≥ | A | for each A ⊂ S 1. Our theorem is that k disjoint matchings of G exist if and only Σ x∈ S 2 min ( k, | F −1( x) ∩ A |) ≥ k | A | for each A ⊂ S 1.